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      SUBROUTINE <a name="DLAED7.1"></a><a href="dlaed7.f.html#DLAED7.1">DLAED7</a>( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q,
     $                   LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR,
     $                   PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK,
     $                   INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N,
     $                   QSIZ, TLVLS
      DOUBLE PRECISION   RHO
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      INTEGER            GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ),
     $                   IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * )
      DOUBLE PRECISION   D( * ), GIVNUM( 2, * ), Q( LDQ, * ),
     $                   QSTORE( * ), WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DLAED7.25"></a><a href="dlaed7.f.html#DLAED7.1">DLAED7</a> computes the updated eigensystem of a diagonal
</span><span class="comment">*</span><span class="comment">  matrix after modification by a rank-one symmetric matrix. This
</span><span class="comment">*</span><span class="comment">  routine is used only for the eigenproblem which requires all
</span><span class="comment">*</span><span class="comment">  eigenvalues and optionally eigenvectors of a dense symmetric matrix
</span><span class="comment">*</span><span class="comment">  that has been reduced to tridiagonal form.  <a name="DLAED1.29"></a><a href="dlaed1.f.html#DLAED1.1">DLAED1</a> handles
</span><span class="comment">*</span><span class="comment">  the case in which all eigenvalues and eigenvectors of a symmetric
</span><span class="comment">*</span><span class="comment">  tridiagonal matrix are desired.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     where Z = Q'u, u is a vector of length N with ones in the
</span><span class="comment">*</span><span class="comment">     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     The eigenvectors of the original matrix are stored in Q, and the
</span><span class="comment">*</span><span class="comment">     eigenvalues are in D.  The algorithm consists of three stages:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        The first stage consists of deflating the size of the problem
</span><span class="comment">*</span><span class="comment">        when there are multiple eigenvalues or if there is a zero in
</span><span class="comment">*</span><span class="comment">        the Z vector.  For each such occurence the dimension of the
</span><span class="comment">*</span><span class="comment">        secular equation problem is reduced by one.  This stage is
</span><span class="comment">*</span><span class="comment">        performed by the routine <a name="DLAED8.45"></a><a href="dlaed8.f.html#DLAED8.1">DLAED8</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        The second stage consists of calculating the updated
</span><span class="comment">*</span><span class="comment">        eigenvalues. This is done by finding the roots of the secular
</span><span class="comment">*</span><span class="comment">        equation via the routine <a name="DLAED4.49"></a><a href="dlaed4.f.html#DLAED4.1">DLAED4</a> (as called by <a name="DLAED9.49"></a><a href="dlaed9.f.html#DLAED9.1">DLAED9</a>).
</span><span class="comment">*</span><span class="comment">        This routine also calculates the eigenvectors of the current
</span><span class="comment">*</span><span class="comment">        problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        The final stage consists of computing the updated eigenvectors
</span><span class="comment">*</span><span class="comment">        directly using the updated eigenvalues.  The eigenvectors for
</span><span class="comment">*</span><span class="comment">        the current problem are multiplied with the eigenvectors from
</span><span class="comment">*</span><span class="comment">        the overall problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ICOMPQ  (input) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  Compute eigenvalues only.
</span><span class="comment">*</span><span class="comment">          = 1:  Compute eigenvectors of original dense symmetric matrix
</span><span class="comment">*</span><span class="comment">                also.  On entry, Q contains the orthogonal matrix used
</span><span class="comment">*</span><span class="comment">                to reduce the original matrix to tridiagonal form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N      (input) INTEGER
</span><span class="comment">*</span><span class="comment">         The dimension of the symmetric tridiagonal matrix.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  QSIZ   (input) INTEGER
</span><span class="comment">*</span><span class="comment">         The dimension of the orthogonal matrix used to reduce
</span><span class="comment">*</span><span class="comment">         the full matrix to tridiagonal form.  QSIZ &gt;= N if ICOMPQ = 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TLVLS  (input) INTEGER
</span><span class="comment">*</span><span class="comment">         The total number of merging levels in the overall divide and
</span><span class="comment">*</span><span class="comment">         conquer tree.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CURLVL (input) INTEGER
</span><span class="comment">*</span><span class="comment">         The current level in the overall merge routine,
</span><span class="comment">*</span><span class="comment">         0 &lt;= CURLVL &lt;= TLVLS.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CURPBM (input) INTEGER
</span><span class="comment">*</span><span class="comment">         The current problem in the current level in the overall
</span><span class="comment">*</span><span class="comment">         merge routine (counting from upper left to lower right).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  D      (input/output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">         On entry, the eigenvalues of the rank-1-perturbed matrix.
</span><span class="comment">*</span><span class="comment">         On exit, the eigenvalues of the repaired matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
</span><span class="comment">*</span><span class="comment">         On entry, the eigenvectors of the rank-1-perturbed matrix.
</span><span class="comment">*</span><span class="comment">         On exit, the eigenvectors of the repaired tridiagonal matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDQ    (input) INTEGER
</span><span class="comment">*</span><span class="comment">         The leading dimension of the array Q.  LDQ &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INDXQ  (output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment">         The permutation which will reintegrate the subproblem just
</span><span class="comment">*</span><span class="comment">         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
</span><span class="comment">*</span><span class="comment">         will be in ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RHO    (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">         The subdiagonal element used to create the rank-1
</span><span class="comment">*</span><span class="comment">         modification.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CUTPNT (input) INTEGER
</span><span class="comment">*</span><span class="comment">         Contains the location of the last eigenvalue in the leading
</span><span class="comment">*</span><span class="comment">         sub-matrix.  min(1,N) &lt;= CUTPNT &lt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1)
</span><span class="comment">*</span><span class="comment">         Stores eigenvectors of submatrices encountered during
</span><span class="comment">*</span><span class="comment">         divide and conquer, packed together. QPTR points to
</span><span class="comment">*</span><span class="comment">         beginning of the submatrices.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  QPTR   (input/output) INTEGER array, dimension (N+2)
</span><span class="comment">*</span><span class="comment">         List of indices pointing to beginning of submatrices stored
</span><span class="comment">*</span><span class="comment">         in QSTORE. The submatrices are numbered starting at the
</span><span class="comment">*</span><span class="comment">         bottom left of the divide and conquer tree, from left to
</span><span class="comment">*</span><span class="comment">         right and bottom to top.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  PRMPTR (input) INTEGER array, dimension (N lg N)
</span><span class="comment">*</span><span class="comment">         Contains a list of pointers which indicate where in PERM a
</span><span class="comment">*</span><span class="comment">         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
</span><span class="comment">*</span><span class="comment">         indicates the size of the permutation and also the size of
</span><span class="comment">*</span><span class="comment">         the full, non-deflated problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  PERM   (input) INTEGER array, dimension (N lg N)
</span><span class="comment">*</span><span class="comment">         Contains the permutations (from deflation and sorting) to be
</span><span class="comment">*</span><span class="comment">         applied to each eigenblock.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  GIVPTR (input) INTEGER array, dimension (N lg N)
</span><span class="comment">*</span><span class="comment">         Contains a list of pointers which indicate where in GIVCOL a
</span><span class="comment">*</span><span class="comment">         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
</span><span class="comment">*</span><span class="comment">         indicates the number of Givens rotations.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  GIVCOL (input) INTEGER array, dimension (2, N lg N)
</span><span class="comment">*</span><span class="comment">         Each pair of numbers indicates a pair of columns to take place
</span><span class="comment">*</span><span class="comment">         in a Givens rotation.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N)
</span><span class="comment">*</span><span class="comment">         Each number indicates the S value to be used in the
</span><span class="comment">*</span><span class="comment">         corresponding Givens rotation.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK   (workspace) DOUBLE PRECISION array, dimension (3*N+QSIZ*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  IWORK  (workspace) INTEGER array, dimension (4*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO   (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  successful exit.
</span><span class="comment">*</span><span class="comment">          &lt; 0:  if INFO = -i, the i-th argument had an illegal value.
</span><span class="comment">*</span><span class="comment">          &gt; 0:  if INFO = 1, an eigenvalue did not converge
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Based on contributions by
</span><span class="comment">*</span><span class="comment">     Jeff Rutter, Computer Science Division, University of California
</span><span class="comment">*</span><span class="comment">     at Berkeley, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP,
     $                   IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           DGEMM, <a name="DLAED8.171"></a><a href="dlaed8.f.html#DLAED8.1">DLAED8</a>, <a name="DLAED9.171"></a><a href="dlaed9.f.html#DLAED9.1">DLAED9</a>, <a name="DLAEDA.171"></a><a href="dlaeda.f.html#DLAEDA.1">DLAEDA</a>, <a name="DLAMRG.171"></a><a href="dlamrg.f.html#DLAMRG.1">DLAMRG</a>, <a name="XERBLA.171"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          MAX, MIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test the input parameters.
</span><span class="comment">*</span><span class="comment">
</span>      INFO = 0
<span class="comment">*</span><span class="comment">
</span>      IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
         INFO = -4
      ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN
         INFO = -12
      END IF
      IF( INFO.NE.0 ) THEN
         CALL <a name="XERBLA.194"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="DLAED7.194"></a><a href="dlaed7.f.html#DLAED7.1">DLAED7</a>'</span>, -INFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      IF( N.EQ.0 )
     $   RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     The following values are for bookkeeping purposes only.  They are
</span><span class="comment">*</span><span class="comment">     integer pointers which indicate the portion of the workspace
</span><span class="comment">*</span><span class="comment">     used by a particular array in <a name="DLAED8.205"></a><a href="dlaed8.f.html#DLAED8.1">DLAED8</a> and <a name="DLAED9.205"></a><a href="dlaed9.f.html#DLAED9.1">DLAED9</a>.
</span><span class="comment">*</span><span class="comment">
</span>      IF( ICOMPQ.EQ.1 ) THEN
         LDQ2 = QSIZ
      ELSE
         LDQ2 = N
      END IF
<span class="comment">*</span><span class="comment">
</span>      IZ = 1
      IDLMDA = IZ + N
      IW = IDLMDA + N
      IQ2 = IW + N
      IS = IQ2 + N*LDQ2
<span class="comment">*</span><span class="comment">
</span>      INDX = 1
      INDXC = INDX + N
      COLTYP = INDXC + N
      INDXP = COLTYP + N
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Form the z-vector which consists of the last row of Q_1 and the
</span><span class="comment">*</span><span class="comment">     first row of Q_2.
</span><span class="comment">*</span><span class="comment">
</span>      PTR = 1 + 2**TLVLS
      DO 10 I = 1, CURLVL - 1
         PTR = PTR + 2**( TLVLS-I )
   10 CONTINUE
      CURR = PTR + CURPBM
      CALL <a name="DLAEDA.232"></a><a href="dlaeda.f.html#DLAEDA.1">DLAEDA</a>( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR,
     $             GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ),
     $             WORK( IZ+N ), INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     When solving the final problem, we no longer need the stored data,
</span><span class="comment">*</span><span class="comment">     so we will overwrite the data from this level onto the previously
</span><span class="comment">*</span><span class="comment">     used storage space.
</span><span class="comment">*</span><span class="comment">
</span>      IF( CURLVL.EQ.TLVLS ) THEN
         QPTR( CURR ) = 1
         PRMPTR( CURR ) = 1
         GIVPTR( CURR ) = 1
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Sort and Deflate eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span>      CALL <a name="DLAED8.248"></a><a href="dlaed8.f.html#DLAED8.1">DLAED8</a>( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT,
     $             WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2,
     $             WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ),
     $             GIVCOL( 1, GIVPTR( CURR ) ),
     $             GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ),
     $             IWORK( INDX ), INFO )
      PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N
      GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Solve Secular Equation.
</span><span class="comment">*</span><span class="comment">
</span>      IF( K.NE.0 ) THEN
         CALL <a name="DLAED9.260"></a><a href="dlaed9.f.html#DLAED9.1">DLAED9</a>( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ),
     $                WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO )
         IF( INFO.NE.0 )
     $      GO TO 30
         IF( ICOMPQ.EQ.1 ) THEN
            CALL DGEMM( <span class="string">'N'</span>, <span class="string">'N'</span>, QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2,
     $                  QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ )
         END IF
         QPTR( CURR+1 ) = QPTR( CURR ) + K**2
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Prepare the INDXQ sorting permutation.
</span><span class="comment">*</span><span class="comment">
</span>         N1 = K
         N2 = N - K
         CALL <a name="DLAMRG.274"></a><a href="dlamrg.f.html#DLAMRG.1">DLAMRG</a>( N1, N2, D, 1, -1, INDXQ )
      ELSE
         QPTR( CURR+1 ) = QPTR( CURR )
         DO 20 I = 1, N
            INDXQ( I ) = I
   20    CONTINUE
      END IF
<span class="comment">*</span><span class="comment">
</span>   30 CONTINUE
      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="DLAED7.285"></a><a href="dlaed7.f.html#DLAED7.1">DLAED7</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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